Nephroid
The two-cusped epicycloid — the bright caustic you see when sunlight reflects inside a cup.
Adjust parameters
Export
Formulas and key values
- Parametric equations
- PerimeterIn the graph = 24
- AreaIn the graph ≈ 37.7
- Cusps
r is the rolling circle’s radius; the fixed circle’s radius is 2r.
Explainer video
Make a video of the Nephroid
Opens a video template with the parameters on screen. Every number in the video is computed before it renders; choose landscape or vertical.
Start the video01 · Definition
What is the Nephroid?
The nephroid — “kidney-shaped” — is the epicycloid with two cusps, traced by a circle rolling around a fixed circle of twice its radius. Proctor named it in 1878, though Huygens had already connected it to optics in 1678.
It is the everyday caustic: when parallel rays (sunlight) reflect off the inside of a circular cup, the reflected rays envelope one half of a nephroid — the bright cusped line on the surface of your coffee. A point source on the rim gives a cardioid instead; the two curves are cousins that answer slightly different lighting setups.
02 · Properties
Key properties
- Arc length 24r and area 12πr².
- It is the epicycloid with k = 2 (R = 2r).
- Caustic of a circle under parallel light — the coffee-cup curve.
- Its evolute is another nephroid, half the size and rotated 90°.
- The name comes from the Greek nephros, “kidney”.
03 · Questions
Nephroid: common questions
- What is the equation of the Nephroid?
- Parametric equations: x = r(3 cos t - cos 3t), y = r(3 sin t - sin 3t). r is the rolling circle’s radius; the fixed circle’s radius is 2r.
- What is the perimeter of the Nephroid?
- Perimeter: L = 24r. r is the rolling circle’s radius; the fixed circle’s radius is 2r.
- What is the area of the Nephroid?
- Area: A = 12π r^2. r is the rolling circle’s radius; the fixed circle’s radius is 2r.
04 · Keep exploring